430 Mr. J. W. L. Glaisher on a Class of Definite Integrals. 



Todhunter remarks (p. 486 of his 'History') that in a memoir 

 of 1783 Laplace pointed out the use of tabulating the integral 

 \e~ t ' dt for different limits. 



In the Theory of the Conduction of Heat the function occupies 

 a place of great importance. Thus Fourier proves* that if one 

 extremity of a bar be kept at a constant temperature equal to 

 unity, the initial temperature at all points in the rest of the bar 

 being zero, then the temperature v at time t will be given by 

 the formula - -" 



c being written for 



vi 



On page 514 of the memoir of Fourier's just referred to is 

 given the proposition from which Sir W. Thomson has deduced 

 his results with regard to the secular cooling of the earth f. 

 Fourier's problem is, that if in a solid extending to infinity in 

 all directions the temperature has initially two different constant 

 values, viz. unity and zero on the two sides of a certain infinite 

 plane, then at time t the temperature v will be given by the 

 equation 



v— --— Lrf =• 



V-k 2\/kt 



Problems dependent for their solution on the error-function 

 are also discussed by Fourier, loc. cit. p. 516 et seqq., and by 

 Riemann, pp. 124, 164, &c. of his Partielle Differentialgleich- 

 ungen, &c. (Braunschweig, 1869). On page 169 Riemann proves % 

 that 



+ ie- 2c |'Erf(«-^-Erf^~|)"|. . . (59) 



A particular case of this elegant theorem was diseovered by 

 Boole § in 1849 ; his result may be written 



* MSmoires de I'Institut, t. iv. (1819 and 1820), p. 508. 

 t Thomson and Tait's * Natural Philosophy,' vol. i. p. 717, where other 

 references are given. 



X Riemann has left out the factor | on the right-hand side. 

 § Cambridge and Dublin Mathematical Journal, vol. iv. p. 18. 



